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verificar (1+cos(t))/(1-cos(t))=(csc(t)+cot(t))^2

Solución

verificar

\frac{1 + cos ( t )}{1 - cos ( t )} = (csc (t) + cot (t))^{2}

Verdadero

Pasos de solución

\frac{1 + cos ( t )}{1 - cos ( t )} = (csc (t) + cot (t))^{2}

Manipular el lado izquierdo

(csc (t) + cot (t))^{2}

Expresar con seno, coseno

(cot (t) + csc (t))^{2}

Utilizar la identidad trigonométrica básica:

cot (x) = \frac{cos ( x )}{sin ( x )}

= (\frac{cos ( t )}{sin ( t )} + csc (t))^{2}

Utilizar la identidad trigonométrica básica:

csc (x) = \frac{1}{sin ( x )}

= (\frac{cos ( t )}{sin ( t )} + \frac{1}{sin ( t )})^{2}

Simplificar

(\frac{cos ( t )}{sin ( t )} + \frac{1}{sin ( t )})^{2} : \frac{( cos ( t ) + 1 ) ^{2}}{sin ^{2} ( t )}

(\frac{cos ( t )}{sin ( t )} + \frac{1}{sin ( t )})^{2}

Combinar las fracciones usando el mínimo común denominador:

\frac{cos ( t ) + 1}{sin ( t )}

Aplicar la regla

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{cos ( t ) + 1}{sin ( t )}

= (\frac{cos ( t ) + 1}{sin ( t )})^{2}

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( cos ( t ) + 1 ) ^{2}}{sin ^{2} ( t )}

= \frac{( cos ( t ) + 1 ) ^{2}}{sin ^{2} ( t )}

= \frac{( 1 + cos ( t ) ) ^{2}}{sin ^{2} ( t )}

Re-escribir usando identidades trigonométricas

\frac{( 1 + cos ( t ) ) ^{2}}{sin ^{2} ( t )}

Utilizar la identidad pitagórica:

cos^{2} (t) + sin^{2} (t) = 1

sin^{2} (t) = 1 - cos^{2} (t)

= \frac{( 1 + cos ( t ) ) ^{2}}{1 - cos ^{2} ( t )}

Factorizar

1 - cos^{2} (t) : (1 + cos (t)) (1 - cos (t))

1 - cos^{2} (t)

Aplicar la siguiente regla para binomios al cuadrado:

t^{2} - y^{2} = (t + y) (t - y)

1 - cos^{2} (t) = (1 + cos (t)) (1 - cos (t))

= (1 + cos (t)) (1 - cos (t))

= \frac{( 1 + cos ( t ) ) ^{2}}{( 1 + cos ( t ) ) ( 1 - cos ( t ) )}

Eliminar los terminos comunes:

cos (t) + 1

= \frac{1 + cos ( t )}{1 - cos ( t )}

= \frac{1 + cos ( t )}{1 - cos ( t )}

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

sec^{2} (α) + csc^{2} (α) = \frac{1}{sin ^{2} ( α ) cos ^{2} ( α )}

sin^{2} (A) + cos^{2} (A) = 1

tan (x) = sin (x) sec (x)

(1 + cot^{2} (θ)) (1 - cos^{2} (θ)) = 1

csc (2 x) + cot (2 x) = cot (x)